Matrix Gamma Distributions and Related Stochastic Processes

There is considerable literature on matrix-variate gamma distributions, also known as Wishart distributions, which are driven by a shape parameter with values in the (Gindikin) set. We provide an extension of this class to the case where the shape parameter may actually take on any positive value. In addition to the well-known singular Wishart as well as non-singular matrix-variate gamma distributions, the proposed class includes new singular matrix-variate distributions, with the shape parameter outside of the Gindikin set. This singular, non-Wishart case is no longer permutation invariant and derivation of its scaling properties requires special care. Among numerous newly established properties of the extended class are group-like relations with respect to the positive shape parameter. The latter provide a natural substitute for the classical convolution properties that are crucial in the study of infinite divisibility. Our results provide further clarification regarding the lack of infinite divisibility of Wishart distributions, a classical observation of Paul Lévy. In particular, we clarify why the row/column vectors in the off-diagonal blocks are infinitely divisible. A class of matrix-variate Laplace distributions arises naturally in this set-up as the distributions of the off-diagonal blocks of random gamma matrices. For the class of Laplace rectangular matrices, we obtain distributional identities that follow from the role they play in the structure of the matrix gamma distributions. We present several elegant and convenient stochastic representations of the discussed classes of matrix-valued distributions. In particular, we show that the matrix-variate gamma distribution is a symmetrization of the triangular Rayleigh distributed matrix { a new class of the matrix variables that naturally extend the classical univariate Rayleigh variables. Finally, a connection of the matrix-variate gamma distributions to matrix-valued Lévy processes of a vector argument is made. Namely, a Lévy process, termed a matrix gamma- Laplace motion, is obtained by the subordination of the triangular Brownian motion of a vector argument to a vector-valued gamma motion of a vector argument. In this context, we introduce a triangular matrix-valued Rayleigh process, which, through symmetrization, leads to a new matrix-variate gamma process. This process when taken at a properly defined one-dimensional argument has the matrix gamma marginal distribution with the shape parameter equal to its argument.